Characterization of surjective partial differential operators on spaces of real analytic functions
Volume 162 / 2004
Abstract
Let $A({\mit \Omega })$ denote the real analytic functions defined on an open set ${\mit \Omega } \subset {{\mathbb R}}^n$. We show that a partial differential operator $P(D)$ with constant coefficients is surjective on $A({\mit \Omega })$ if and only if for any relatively compact open $\omega \subset {\mit \Omega }$, $P(D)$ admits (shifted) hyperfunction elementary solutions on ${\mit \Omega }$ which are real analytic on $\omega $ (and if the equation $P(D)f = g$, $g\in A({\mit \Omega })$, may be solved on $\omega $). The latter condition is redundant if the elementary solutions are defined on $\mathop {\rm conv}\nolimits ({\mit \Omega })$. This extends and improves previous results of Andersson, Kawai, Kaneko and Zampieri. For convex ${\mit \Omega }$, a different characterization of surjective operators $P(D)$ on $A({\mit \Omega })$ was given by Hörmander using a Phragmén–Lindelöf type condition, which cannot be extended to the case of noncovex ${\mit \Omega }$. The paper is based on a surjectivity criterion for exact sequences of projective (DFS)-spectra which improves earlier results of Braun and Vogt, and Frerick and Wengenroth.
